This question already has an answer here:
At the moment different paths (by percentual very few people in the world) are taken to arrive (that is, if an arrival exists) at a theory that can quantise the curvature of spacetime. Considering the three basic forces that operate in spacetime I can imagine them to be quantized (and which obviously has been done in the three gauge field theories of those forces) in spacetime, but what is the argument that says that also the curved spacetime itself is subject to quantization?
It is clear that a curved spacetime, wich lets masses (in the absence of the e.m. strong- and weak force) interact without acceleration (in the sense of a force), is very different from the stuff out off which the workings of three real basic forces are made.
The search is on for many years now, but a solution hasn´t emerged. Can this be a sign of a wild goose chase or is the assumption an axiom, in which case there is nothing left to be answered?